The idea of TOPSIS can be expressed in a series of steps.
(1) Obtain performance data for n alternatives over k criteria. Raw measurements are usually standardized, converting raw measures xij into standardized measures sij .
(2) Develop a set of importance weights wk, for each of the criteria. The basis for these weights can be anything, but, usually, is ad hoc re°ective of relative importance. Scale is not an issue if standardizing was accomplished in Step 1.
(3) Identify the ideal alternative (extreme performance on each criterion) s+;
(4) Identify the nadir alternative (reverse extreme performance on each criterion) s-;
(5) Develop a distance measure over each criterion to both ideal (D+) and nadir (D-).
(6) For each alternative, determine a ratio R equal to the distance to the nadir divided by the sum of the distance to the nadir and the distance to the ideal, R = D-/(D+ + D-)
(7) Rank order alternatives by maximizing the ratio in Step 6.
Thus, TOPSIS minimizes the distance to the ideal alternative while maximizing the distance to the nadir. There are a number of specific procedures that can be used for Step 2 (developing weights), and for Step 5 (distance measures). Additionally, different conventions can be applied to defining best performance (Step 3) and worst performance (Step 4).
A number of distance metrics can be applied. Traditional TOPSIS applied the Euclidean norm (minimization of square root of the sum of squared distances) to ideal and nadir solutions, a second power metric (P2). TOPSIS2 is a variant where distance was measured in least absolute value terms, a first power metric (P1). Another commonly used metric is the Tchebychev metric, where the minimum maximum difference is the basis for selection. This coincides with an infinite power-term (P1).